Abstract There are two ways to study quantum liquids. One is the fully microscopic treatment, which can be realised either by numerical simulations of the many-body problem, or for some special ranges of the material parameters, analytically. The other is a phenomenological approach in terms of effective theories. The hierarchy of the effective theories corresponds to the low-frequency, long-wavelength dynamics of quantum liquids in different ranges of frequency. This chapter discusses the microscopic (atomic) physics of quantum liquids which is the analog of the Theory of Everything — the Planck scale physics, difference between microscopic particles and emerging quasiparticles, the interplay of microscopic and effective symmetries, the origin of fundamental constants and hierarchy of Planck energy scales in the ‘Theory of Everything’, vacuum pressure and vacuum energy and scenario of vanishing of cosmological constant. These are considered using the model of weakly interacting Bose gas, the model of quantum liquid, and the real liquid 4He.
Abstract There are two ways to study quantum liquids. One is the fully microscopic treatment, which can be realised either by numerical simulations of the many-body problem, or for some special ranges of the material parameters, analytically. The other is a phenomenological approach in terms of effective theories. The hierarchy of the effective theories corresponds to the low-frequency, long-wavelength dynamics of quantum liquids in different ranges of frequency. This chapter discusses the microscopic (atomic) physics of quantum liquids which is the analog of the Theory of Everything — the Planck scale physics, difference between microscopic particles and emerging quasiparticles, the interplay of microscopic and effective symmetries, the origin of fundamental constants and hierarchy of Planck energy scales in the ‘Theory of Everything’, vacuum pressure and vacuum energy and scenario of vanishing of cosmological constant. These are considered using the model of weakly interacting Bose gas, the model of quantum liquid, and the real liquid 4He.
Abstract This chapter deals with the Fermi systems, where the low-energy effective theory involves both bosonic and fermionic fields. Above the phase transition to the superconducting or superfluid state, the overwhelming majority of systems consisting of fermionic particles (electrons in metals, neutrons in neutron stars, 3He atoms in 3He liquid, etc.) form a so-called Fermi liquid. Below transition new types of fermionic vacua emerge. This chapter discusses the Bardeen–Cooper–Schrieffer (BCS) theory for spin-triplet superfluids, which provide examples of different universality classes of fermionic vacua: fully gapped vacua, vacua with stable and marginal point nodes — Fermi points, and vacua with nodal lines — Fermi lines. It also discusses emergent ‘relativistic’ quasiparticles, fundamental constants and hierarchy of Planck energy scales in fermionic systems, problem of vacuum energy and cosmological term in bi-metric gravity, and mass generation for Standard Model fermions.
The macroscopic theory of Fermi liquids we have developed in the preceding chapters possesses the advantage of being exact for any normal liquid, irrespective of its density or law of interaction, so long as the temperature is sufficiently low and the phenomena under discussion involve both long wavelengths and low frequencies. It is, of course, incomplete, since it can provide no account of microscopic phenomena involving short wavelengths and large excitation energies, or of equilibrium properties such as the ground state energy and pair distribution function. In contrast, microscopic theories are able to describe both macroscopic and microscopic phenomena. They are, however, subject to one of the following limitations: They are exact for only a limited range of particle density or interaction, one which rarely corresponds to physical reality. The extent to which they provide an approximate account of a real physical system can rarely be estimated with any precision.
We conclude our study of superfluidity in Bose liquids with a brief summary of the strengths, and of the inadequacies, of the theory. The cornerstone of our fundamental understanding of superfluidity has been the recognition of the central importance of the condensate, a single quantum state which is macroscopically occupied. For the Bose liquid one has a particle condensate, characterized by a non-vanishing expectation value of ψ (r). The existence of the condensate explains all major aspects of superfluid behavior; in particular, we now possess: A clear understanding of the static and the long wavelength dynamic properties of an equilibrium Bose liquid, including a microscopic basis for the two-fluid model. A precise microscopic theory at T = 0 for a specific model, the dilute Bose gas. Variational calculations of the excitation spectrum of liquid He II (and of the ground state properties as well), which are in good quantitative agreement with experiment. A satisfactory phenomenological treatment of the interactions between elementary excitations which give rise to various irreversible phenomena. A qualitative understanding of the role which is played by quantized vortices in both equilibrium and non-equilibrium phenomena.
This chapter concludes the study of superfluidity in Bose liquids with a brief summary of the strengths, and of the inadequacies, of the theory. A clear understanding of the static and the long wavelength dynamic properties of an equilibrium Bose liquid, including a microscopic basis for the two-fluid model. Variational calculations of the excitation spectrum of liquid He II, which are in good quantitative agreement with experiment. A satisfactory phenomenological treatment of the interactions between elementary excitations which give rise to various irreversible phenomena. A microscopic theory of the ground state and elementary excitation spectrum of liquid He II; in fact, even the theory of the simple dilute Bose gas has not yet been extended to treat the system at finite temperatures, and under non-equilibrium situations. A clear understanding of a number of "vortex" properties, including the critical current for superfluid flow in thin capillaries.
Envisioned by Richard Feynman in the early 1980s, quantum simulation has received dramatic impetus thanks to the development of a variety of plateforms able to emulate a wide class of quantum Hamiltonians.During the past decade, most of the quantum simulators have implemented rather well-known models, hence permitting a direct comparison with theoretical calculations and a precise benchmarking of their reliability.The field has now reached a maturity such that one can address difficult problems, which cannot be solved efficiently using classical algorithms.These advances provide unprecedented opportunities to explore previously unreachable fields, test theoretical predictions, and inspire novel approaches.This contribution is an elementary introduction to quantum simulation.We discuss the challenges, define both digital and analog quantum simulators, and list the demanding conditions they require.We also provide a brief account of the contributions gathered in the dossier on Quantum simulation in the Compte rendus Physique of the French Academy of Sciences [1][2][3][4][5][6].The latter completes excellent reviews that appeared previously, see for instance . Universal models and the role of simulations in many-body physicsUnderstanding the behavior of macroscopic quantum systems is a major challenge of modern physics.The basic laws of low-energy physics are by now quite well known at the microscopic, say atomic, scale.Conversely, many fundamental questions remain open, and even debated, about the collective dynamics at the macroscopic scale.By "macroscopic scale", here we mean systems made up of a huge number of constituents, or degrees of freedom, say 10 6 , 10 23 , or even more, as relevant in condensed matter physics or in astrophysics, for instance.Such huge systems cannot be treated exactly, be it a the classical level and, even worse, at the quantum level.Yet, it is the main outcome of the thermodynamic approach that the collective behavior of a macroscopic system can drastically differ from that of its elementary constituents.For instance, the elementary interactions between the H 2 O molecules are fundamentally unchanged when a water bucket turns from the liquid phase to the solid phase at zero degree Celsius.Similarly, the interactions between the microscopic magnetic moments do not show any brutal change when a magnetic material gets magnetized underneath the Curie temperature.Hence, the dramatic effects observed in macroscopic systems are governed by large-scale instabilities, without obvious counterparts at the microscopic level.This observation takes a universal character, summarized in the celebrated motto "More is Different" [15].Such so-called emerging phenomena also appear in quantum systems, where new effects that are impossible in the classical world show up below some critical temperature or at zero temperature when some interaction parameter passes through a critical value.Celebrated examples include the superfluid transition in helium or super-conducting transitions and other metal-insulator transitions in electronic systems.Strikingly enough at first sight, while emerging phenomena only appear in very large scale macroscopic systems, their germs are contained in the mutual interactions of their elementary constituents.Hence, local two-body interactions are sufficient to explain a huge class of phase transitions, such as the liquid-solid and magnetic transitions mentioned above.The Boltzmann statistical approach proved particularly successful in describing this connection between the microscopic and macroscopic scales.To understand emerging phenomena on grounds as fundamental as possible, the programme is now well established: One tries to identify the basic microscopic terms that seem to be relevant and disregards all the other microscopic details.One then elaborates a model, as generic as possible, likely to reproduce the main experimental observations.The most usual examples are the Ising and Heisenberg models for magnetic transitions, or the Hubbard model for metal-insulator transitions [16][17][18].Then all is left to do is to check that the phenomenon of interest indeed emerges from the dynamics of the simplified model.The realization of this programme is nothing but a simulation.It consists in building up a simplified system that mimics the main properties of a real system.
Abstract In this chapter we outline the microscopic theory of an interacting Bose fluid. The aim is to provide a basis for a microscopic understanding of the excitations in superfluid and normal 4He discussed in the following chapters. The microscopic picture can be omitted on a first reading and referred to when desired.We assume that the reader is familiar with the theory of Bose-Einstein condensation in a Bose gas introduced in §7.2 and presented in many texts on statistical mechanics. In §7.2 we also discussed the generalization of Bose condensation to an interacting fluid by Penrose and Onsager (1956).Condensation was identified with macroscopic occupation of one of the eigenstates of the fluid and the appearance of an infinitely long tail in the one-body density matrix, denoted as off-diagonal long range order (ODLRO). For a homogeneous fluid it is natural to take momentum states as the eigenstates and to assume that the macroscopically occupied state is the state with momentum p = 0. In this case the criterion reduces to that for a gas but with a firm foundation.